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This question might be too conceptual.
Congruences between modular forms (due to Shimura, Hida, etc) are really amazing. I know that the eigencurve construction are closely related to these relations. The basic reference is "The Eigencurve" by Coleman and Mazur. Besides, I think "A brief introduction to the work of Haruzo Hida" by Mazur is a good introduction.

It seems that we at first prove the congruence and then interpolate them into a family, like the Eisenstein family, and the property of the eigencurve are deduced from properties of modular forms.

So I wonder conversely, can the eigencurve construction explain (prove) more congruences between modular forms (not just these used in building the eigencurve)? Or other interesting facts about modular forms? For example, see section 5 of "A brief introduction to the work of Haruzo Hida".

I type slowly, sorry...

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    $\begingroup$ Please improve (clarify) your question by adding more detail, context, and references. $\endgroup$
    – GH from MO
    Commented Oct 15, 2015 at 1:46

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Here's a theorem about congruences between modular forms:

Theorem. Let $f$ be a (normalised) eigenform of weight $k$ and level $\Gamma = \Gamma_1(N) \cap \Gamma_0(p)$. Then for any $r$, and any $k'$ sufficiently close (*) to $k$, there exists an eigenform of weight $k'$ and level $\Gamma$ that is congruent to $f$ modulo $p^r$.

This is an incredibly powerful theorem; it's virtually self-evident once you know the eigencurve exists; and I don't think I know of any way of proving it without constructing the eigencurve in the process.

Does that answer your question?

(*) Here "close" means that $k'$ has to be congruent to $k$ modulo $(p-1) p^j$, for some $j$ depending on $f$, $p$ and $r$.

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  • $\begingroup$ Thanks, David. It seems that the eignnvariety machine really visualizes properties of modular forms. But this result seems different from the results I know (basically Shimura or Hida's). So is it possible to deduce Shimura's (In A tameness criterion for Galois representations associated to modular forms (mod p) by Gross proposition 9.3 p. 478., Between weight k and weight 2) or Hida's results (between modular forms of same wight I think)? $\endgroup$
    – user42690
    Commented Oct 16, 2015 at 2:30
  • $\begingroup$ And is there no way prove this theorem without constructing the eigencurve? At least if $j=0$ it's just multiplying with the Hasse invariant (Eisenstein series I mean). Maybe I shall post another MO question...... $\endgroup$
    – user42690
    Commented Oct 16, 2015 at 3:51
  • $\begingroup$ No, it is much more powerful than just multiplying by Eisenstein series, because it's showing that there are eigenforms of weight $k'$ highly congruent to $f$, not just any old forms of that weight. $\endgroup$
    – David Loeffler
    Commented Oct 16, 2015 at 6:18

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